https://mathresearch.utsa.edu/wiki/index.php?action=history&feed=atom&title=Continuous_Vector_Functions
Continuous Vector Functions - Revision history
2026-04-29T05:36:50Z
Revision history for this page on the wiki
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https://mathresearch.utsa.edu/wiki/index.php?title=Continuous_Vector_Functions&diff=3278&oldid=prev
Khanh at 15:28, 30 October 2021
2021-10-30T15:28:04Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 15:28, 30 October 2021</td>
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<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>http://mathonline.wikidot.com/continuity-of-vector-valued-functions</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">Like with continuity of single variable real-valued functions, the continuity of vector-valued functions is as follows:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"><blockquote style="background: white; border: 1px solid black; padding: 1em;"> </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">: '''Definition:''' Let <math>\vec{r}(t)</math> be a vector-valued function. We say that <math>\vec{r}(t) = (x(t), y(t), z(t))</math> is '''Continuous at <math>a</math>''' if <math>\lim_{t \to a} \vec{r}(t) = \vec{r}(a)</math>, that is <math>\lim_{t \to a} x(t) = x(a)</math>, <math>\lim_{t \to a} y(t) = y(a)</math> and <math>\lim_{t \to a} z(t) = z(a)</math>. We say that <math>\vec{r}(t)</math> is '''Continuous on The Interval <math>I</math>''' if <math>\vec{r}(t)</math> is continuous for all <math>t \in I</math>.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"></blockquote></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">We should note that from the definition of <math>\vec{r}(t)</math> being continuous at <math>a</math>, we must have that the component functions <math>x(t)</math>, <math>y(t)</math>, and <math>z(t)</math> all be continuous at <math>a</math>.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">For example, suppose we wanted to show that <math>\vec{r}(t) = (t, -2t^2, \cos t)</math> is continuous at <math>\pi</math>. We notice that <math>x(t) = t</math>, <math>y(t) = -2t^2</math>, and <math>z(t) = \cos t</math> are all continuous real-valued functions, and so they're also all continuous at <math>\pi</math>. Therefore, we have that:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"><div style="text-align: center;"><math> \begin{align} \quad \quad \lim_{t \to \pi} \vec{r}(t) = \left ( \lim_{t \to \pi} t, \lim_{t \to \pi} -2t^2, \lim_{t \to \pi} \cos t \right ) = (\pi, -2\pi^2, -1) = \vec{r}(\pi) \end{align}</math></div></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">==Licensing==</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">Content obtained and/or adapted from:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">* [</ins>http://mathonline.wikidot.com/continuity-of-vector-valued-functions <ins class="diffchange diffchange-inline">Continuity of Vector-Valued Functions, mathonline.wikidot.com] under a CC BY-SA license</ins></div></td></tr>
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Khanh
https://mathresearch.utsa.edu/wiki/index.php?title=Continuous_Vector_Functions&diff=3177&oldid=prev
Lila: Created page with "http://mathonline.wikidot.com/continuity-of-vector-valued-functions"
2021-10-28T22:11:02Z
<p>Created page with "http://mathonline.wikidot.com/continuity-of-vector-valued-functions"</p>
<p><b>New page</b></p><div>http://mathonline.wikidot.com/continuity-of-vector-valued-functions</div>
Lila